Some enumerations on non-decreasing Dyck paths

Abstract

We construct a formal power series on several variables that encodes many statistics on non-decreasing Dyck paths. In particular, we use this formal power series to count peaks, pyramid weights, and indexed sums of pyramid weights for all non-decreasing Dyck paths of length 2n. We also show that an indexed sum on pyramid weights depends only on the size and maximum element of the indexing set.